STRONGLY POINTLIKE MAPS AND COLLAPSEBLE COMPLEXES Ykests he Hm an?“ a5 pk. D. WEE-KERN STATE UREVERSI'EY Pm: Francis Eli-ice? am: 1988. LIBRARY Michigan State University This is to certify that the thesis entitled STRONGLY POINTLIKE MAPS AND COLLAPSIBLE COMPLEXES presented by Paul Francis Dierker has been accepted towards fulfillment of the requirements for Major profess I Date AuguSt 19, 1966 0-169 ABSTRACT STRONGLY POINTLIKE MAPS AND COLLAPSIBLE COMPLEXES by Paul Francis Dierker Let [aiii = 1,2,...,s} be the set of barycenters of the simplexes in the simplicial complex L. If leKI 9E£2> [Li is a simplicial map with the property that f;1(ai) is collapsible for all i = 1, o.., s, we call f a strongly pointlike map. In this thesis we examine the relation that exists be- tween a complex and its image under a strongly pointlike map. In particular, in Section I we find that a complex and its image under such a map must be of the same simple homotopy type. Next, collapsible polyhedra are characterized as those polyhedra having a strongly pointlike map onto a 1-simplex. This characterization is then used to derive a few prOperties of collapsible polyhedra. In Section V the characterization is used to find a class of non-collapsible polyhedra whose product with the unit interval is collapsible. STRONGLY POINTLIKE MAPS AND COLLAPSIBLE COMPLEXES BY Paul Francis Dierker A THESIS I Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1966 'l‘r l / /' / ‘:t I ' ‘ , j " , . . v" ”’ v ” ,4 I V R. ”ma/L: / ACKNOWLEDGMENTS I am indebted to Professor P. H. Doyle for his helpful guidance during the preparation of this thesis. ii DEDICATION To my wife . iii SECTION I. II. III. IV. CONTENTS INTRODUCTION . . . . . . . . . . . . . . . STRONGLY POINTLIKE MAPS AND SIMPLE HOMOTOPY TYPE . . . . . . . . . . . . . . STRONGLY POINTLIKE MAPS ONTO COLLAPSIBLE COMPLEXES--A CHARACTERIZATION OF COLLAPSIBLE COMPLEXES . . . . . . . . . . . . . . . . FURTHER RESULTS CONCERNING COLLAPSIBLE COMPLEXES . . . . . . . . . . . . . . . . CONTRACTIBLE COMPLEXES WHOSE PRODUCT WITH THE UNIT INTERVAL IS COLLAPSIBLE . . . . . BIBLIOGRAPHY . . . . . . . . . . . . . . . iv Page 20 28 39 SECTION I INTRODUCTION Throughout this thesis we will be considering both simplicial complexes and convex linear cell complexes. All complexes (simplicial or convex linear cell) are to be fin- ite and embedded in some Euclidean space. If K is a complex,[K| will be used to denote the underlying point set of K, that is, the union of all convex linear cells of K. If K is a convex linear cell complex and T a simplicial complex with [TI = |K[, T will be called a simplicial division of |K|. Bn will be called a polyhedral n-cell if there is a piecewise linear homeomorphism from the standard n-simplex onto Bn. If K1 and K2 are complexes with K1 Z>K2 we say that there is an elementary simplicial collapse from K1 to K2 if -K1 = K2 U (a * on) and K2 0 (a * on) = a * 5n. K1 simplicially collapses to K2 (written [K11 \§§~[K2D if there is a sequence of elementary simplicial collapses from K1 to K2. If K2 consists of a single vertex,K1 is said to be simplicially collapsible (written [K1| ‘;\\O). It is well known [3] that if |K1| ‘§m\0,then K1 may be collapsed to any of its vertices. 2 If [K1] :>[K2[ we say that there is an elementary col- lapse from [K1| to [Kzl if there exist Bn and Bn-l, polyhedral 1 n and n-1 balls respectively, with Em” C fin, such that |K1| = |K2| U Bn and [KZI n B“ = Bi‘l [K1] collapses to [K2] (written [K1|‘\§~[K2|) if there is a sequence of elementary collapses from [Kll to 1K2}. If [K2] is a point,|K1| is said to be collapsible (written [K1[\\\.0). The following well known theorem [S] gives the connection between collapsible and simplicially collapsible polyhedra. If [K|‘\§.O then there is a simplicial division T of K such that |T| \;§.0. Let K and L be complexes. We say that K and L have the same simple homotopy type if there exists a complex P such that [PI \ [K| and [P[ \ |L Let L be a subcomplex of the simplicial complex K. The star of L in K is defined as u[ceK|cn[L|#¢} l'." I and the link of L in K as 1km) If 0 is a simplex of K the reduced star of o in K is defined um e Kl CEstK(L); cm [L[ = e} as ;;K(o) - U [C e KlO < C} A subcomplex L of a simplicial complex K is full in K if any simplex of K whose vertices belong to L is in L. In addition we will consistently use A to denote the boundary ova and int A to denote the interior of A. SECTION II STRONGLY POINTLIKE MAPS AND SIMPLE HOMOTOPY TYPE Definition 2.1: Let Vf:|K| > [Ll be a simplicial map of the complex K onto the complex L, and let .[aili = 1,2,...,S} be the set of 'barycenters of the simplexes of L. If f-1(ai) is collapsible for all i the map f is said to be strongly pointlike. In this section we will show that the existence of a strongly pointlike map between two polyhedra implies that the polyhedra are of,the same simple hOmotopy type. This result will follow quickly from [1]. Definition 2.2: If M is a subcomplex of the complex 1 . . +1 . K C En CEn+ and V 15 a pOint of En - En we define the quotient complex of K with respect to M as K/M = [K - K|stK(M)]U[v * Klle(M)]. Lemma 2.1: Let M be a collapsible subcomplex of K and suppose that |K|stK(M)| \;n\M. Then.[K/M[ is of the same Simple homotoPy type as [Kl. Proof: Since a cone collapses simplicially to any subcone,[5] we have that [v * KlstK(M)[ ‘;\\[v * Klle(M)(. Thus [v * KlstK(M) U [K - KlstK(M)][ ';\\[K/M[. Moreover [K[stK(M)[ ';\\[M[ \\§ 0 so [KlstK(M)[‘\\\ 0. Thus by lemma 3 of [3] 3 4 [v u» KlstK(M) U [K - KlstK(M)]l \ lKl and so lK/Ml and lKl are of the same simple homotopy type. Definition 2.3: A subcomplex L of a complex K is locally collapsible if for any simplex o of K, stK(o)fllLl is collapsible. The following lemma is the second portion of proposition 2.1 of [1]. Lemma 2.2: Let M be a subcomplex of K such that (i) M is full in K (ii) M is locally collapsible in K, then lKlstK(M)l ‘gmklMl. Definition 2.4: The simplicial mapping sz -—> K/M defined by Fl lKl - stK(M) = id and w(lMl) = V is called the projection of -K onto K/M. onto) l Theorem 2.3: If leKl Ll is a strongly point- like map,then [Kl and lLl have the same simple homotopy type. .Proof: Order thesimplexes of L. in the order of in- creasing dimension, §1,C2,...,CS and let ai denote the barycenter of Ci. Construct the sequence of quotient com- plexes n u H K0 K1 K2 W -1 _i N Ks where L' is the first barycentric subdivision of L and K' is a first derived complex of K chosen so that f:K' > L' is simplicial [1]. From [1] we have that KS is isomorphic to L', and -1 . that Ji - Ki_1lwi_1...w1f (ai) is both full and locally collapsible in Ki-1' Thus by Lemma 2.2 lKi-ilStKi_1(Ji)l ‘;‘\lJil ° In addition, since f is strongly pointlike -1 . . . f (ai)\\§\0 for all 1.. It 18 an easy calculation to show -1 -1 -1 . . that f (ai) fl stK(f (aj)) C le(f (aj)) for all j # l. o" . . -1 Thus wi-1’Fi-2"'°’W1 are all the identity on f (ai) and so [Jil = Wi_1 Vi-2 ...w1f-{(ai)\\m\0. Now we may apply Lemma 2.1 to see that lKil = is of the same simple homotopy type as Ki_1l. Thus [Ksl is of the same simple homotopy type as lK'l, and so [K'l and lLfl are of the same.simple homo- tOpy type. SECTION III STRONGLY POINTLIKE MAPS ONTO COLLAPSIBLE COMPLEXES A CHARACTERIZATION OF COLLAPSIBLE COMPLEXES In this section we will establish that the preimage of a collapsible polyhedron under a strongly pointlike map is collapsible. In order to establish this it is helpful to prove first that the preimage of a one simplex under a strongly pointlike map is collapsible. Both the more general result and a characterization of collapsible com— plexes will follow from the consideration of this special case. The idea for the proof of the special case is quite simple and is indicated schematically below. Let f: lKl onto (V0,V1> be strongly pointlike, b denote the barycenter of (V0,V1>, and b e (bo,b1> C int. Then lKl may be represented pictorially as in Figure 1. By following the path of collapsing f-1(b) to p we will show that K may be collapsed to the complex repre- sented pictorially in Figure 2. This in turn will be col- lapsed to the one cell (U0,U1> which is collapsible. Remark: Let f:on 92E2> (V0,V1> be a simplicial map. Further let b denote the barycenter of (V0,V1> and (b1,b2> a one cell contained in (V0,V1>. Then since f is a linear map we have n-1 f-1(b) = B , a convex linear (n-1)-cell f-1 = Bn, a convex linear n-cell 6 Figure 1. Figure 2. —1 (f‘1f = [f’1(bo) U f (b1)1 u [f‘1né“1. We shall use these facts throughout the proofs of the fol- lowing lemmas. onto Lemma 3.1: Let f: Kl -———> (V0,V1> be a simplicial map, b the barycenter of (VO,V1>, and (bo,b1>cint. There is a piecewise-linear homeomorphism h:f-1(b) x I 93E9> f'1 such that h(f-1(b) x {0}) = f-1(bo), h(f-1(b) x {1}) = f-1(b1), and h((f’1(b)no) x I) = f’1 n o, o e K. Egggg: The proof will be by induction on k , the number of simplexes o e K with o n f—1(b) # ¢. If k = 1 and o n f-1(b) # ¢ then 0 must be a one simplex. Other- wise 0 fl f-1(b) # ¢ implies. 6 fl f-1(b) # ¢. Then there is another simplex C C 6 such that C O f-1(b) # ¢ and so k # 1. Thus if k = 1 ,f-1(b) is a point and f_1(b) x I a polyhedral one-cell. f-1 is also a polyhedral one— cell with boundary f_1(bo) U f_1(b1). The conclusion of the theorem is then obvious for k = 1. Now suppose that the lemma is true for k and let K have k + 1 simplexes with o n f_1(b) # ¢. Let C be a principal simplex of K with C 0 f-1(b) # ¢. Then K - C onto is a simplicial complex and leK — Cl > (VO,V1> a simplicial map. By the induction hypothesis there is a 9 piecewise linear homeomorphism onto g:[f-1(b) n lK - all x I > f_1 n lK - Cl with 9([f'1(b) n [K - all x {0}) f-1(bo) n lK — cl g<[f'1 n O. o e K f-1(b1) n lK - cl and Note that i c lK - Cl so g is defined on (f-1(b) n i) x I. Moreover, 'g[(f_1(b) 0 i) x I] = f'1 n i so guf‘lao) n i) x (on g[(f’1(b) n 2:) x {1}] (f-1(b) n i) x {0] and (f_1(b) H i) x {1] are the boundaries f-1(bo) n i * and f-1(b1) n i. of the polyhedral balls (f_1(b) n c) x [0} and (fm1 (b) H C) X {1] respectively. In addition, f-1(bo)n & f-1 (b1) 0 é are the boundaries of the polyhedral balls f-1(b0) n c and f-1(b1) n g respectively. By [5] any piecewise linear homeomorphism between the boundaries of two polyhedral balls can be extended to the interiors. Thus we may extend g to g) a piecewise linear homeomorphism with 'g'Hf‘lao) n C) x {0}] 3[ (f‘1(b> n C) x {1}] Now 5' is defined on f-1(bo) n c and f-1(b1) n c . s = t(f‘1x{01JUI(f‘laommxmwl(f‘1(b)n5:)x11 . the boundary of the polyhedral ball (f-1(b) n C) x I, and 615) = (f_1(b0) n c) u (f_1(b1) n g) u (f‘1 mi), —1 the boundary of the polyhedral ball f (bo,b1> n C. Thus, 10 as before, we can extend 6' to a piecewise linear homeo- morphism h with h[ n c. Note that since h is an extension of '3 1(b0) n c and Ian) 0 c. :[f-1(b) n lKl] x I 9359> f'1 n [Kl h[(f'1(b) n C) x {0}] h[(f-1(b) n C) x [1)] so h is the desired homeomorphism. Lemma 3.2: Let f:K ——> (V0,V1> be a simplicial map and C int I then f-1 \ f-1(Vo) and f‘1‘\s.f'1(v1). Egggg: 'We will show that f‘1‘\s\f‘1(vo) by induction on k , the number of simplexes of K with 1(b0) n o # e. If k = 1, then as in the previous lemma 0 is a one simplex and f-1 n o a polyhedral one cell with a free vertex f_1(bo). Thus f'1-[int(f"1u 0) U f"1(b0)], and since k = 1, . . f-1-[int(f_1 n o) u f-1(bo)] = f‘1(vo). Now assume that the lemma is true for k and suppose that there are k+1 simplexes in K with o n f-1(b0) # ¢. Let fin be a principal simplex of K‘ with fin n f-1(bo) # ¢. f-1 n fin is a convex linear n-cell with a face {F71(bo) n fin. We now show that we can collapse f 1nCn across f—1(bo) n Cn 11 Since fin is a principal simplex of K , lK-CnlflCn = in so (f‘1 n lK-Cnl) n (f‘1 n g“) = f'1 n in . Moreover f-1 n in is a polyhedral (n-1) cell since f‘1n&n=[(f‘1nén)u n c“)' - \\\ f (Vo,bo> n [K - cl and employ the induction hypothesis to find f1 n lK " Cl\f1)(Vo The proof that f 1\\\ f 1(V1) is analogous. Lemma 3.3: If P\\§.O then PXI \\\(PX[O})U(pxI)U(Px[1]) where p e P. Proof: Let P = Po\\\.Pf\\x °-°\\\.Pi\‘xpi+:\\ ..>\\P£\\p be a sequence of elementary collapses. That is, .n B? 0 P. = Bp-l where B? and BP-1 1 1+1 1 l 1 are polyhedral n and n-1 balls respectively with Bg—l C DE. Thus B? X I is a polyhedral (n+1) ball and -1 . by [51,(32 x I) U (B? x {0}) U (B? x {1}) CI(BE x I) is a polyhedral n-ball since BE-l x I, B? x [0], and 12 B? x [1) are all polyhedral n—balls and (B? x {0}) U (BE-1 X 1» n—L and (B? x [1}) 0 (Bi x I)' are polyhedral. (n-l) balls. Moreover Pi x I = [(1>i+1 x I) U (B? x {0}) U (B? x {1})] U (B? x I) and [(Pi+1 x I) U (B? x {0}) U (B? x {1))] n (B? x I) = (1312"1 x I) U (B? x [0]) U (B? x {1}) SO Pi x I\\(Pi+1 x I) U (B? x {0}) U (B? x [1]) for all i. Then P x I \\\(P1 x I) U (B0 x [0]) U (Bo x (If) \wz x I) U [(8. U8.) ><{0}1 UHBO‘ UB1)>< {111 -1 ' k-I \(pk x I) U U]; Bj x {0}] u [ujd Bj x (1)] \(pXI) U (Px {01) U (Px {1}) Theorem 3.4: If .f:|K| 22329 (V0,V1> is a strongly pointlike map,then |K|\\\_O. Proof: If C int (V0,V1>, then -1 -1 -1 IKI f U f U f - By Lemma 3.1 there is a piecewise linear homeomorphism -1 ' _ _, h:f (b) x I 9939> f 1 such that h(f 1(b) x {0}) ._ .... 7 -1 - = f 1(b0) and h(f 1(b) x {1}) = f (b1). By Lemma 3.3 and the hypothesis that f-1(b)\\\.0 we have f_1(b)XI\\\(f-1(b)x[0])U4pXI)U(f-1(b)x{1}). p e f‘1(b). 13 Let 1(b)xr\xtP;\e...?\e.Pk\\t(f‘1U(f'1(b)x[1}) be a sequence of elementary collapses. Then since h is a piecewise linear homeomorphism ‘ f_10,b1>\\:=\f"1 (b0) U h(p x I) U f (b1). Since f-1, f_1 and f-1 intersect only on 1 _ f (be) and f 1(b1), we have _1 -1 K \\\f' U h(p X I) U f (b1,V1>. Note also that h(p x I) n f‘1 = h(p x {0}) and h(p x I) n f;1 - h(p x {1}). From Lemma 3.2 we have f-1 and by hypothesis _f-1\\\0‘ so f-1\\\0. -By [5] there is a simplicial subdivision T of f—1- such that T ‘;\\0 and h(p x [0]) is a vertex of T. (Star from h(p x {0}) if necessary.) Then we may collapse T simplicially to h(p x {0}). Similarly f-1 \h(p ><[1}) . Thus K \f U h(p x I) U f _\h(p x I)\0 and the theorem is proved. In order to prove a similar theorem in the general case where the range is collapsible, and not necessarily a one simplex, we need the following two trivial lemmas. 14 Lemma 3.5: Let C 1 and 52 be n-dimensional convex linear cell complexes. Suppose there exists a map 9: C1 2%E%> € 2 such that (a) g(AnB) = g(A) fl g(B) for all A,B 6 Cl (b) dim g(A) = dim A for all A e C 1. onto Then there is a piecewise linear homeomorphism h: I C1| >- onto [CZI such that h: A] > |g(A)| for all A661. Proof: The proof will be by induction on k , the cardinality of C1. If k = 1 the result is clear. (In this case I51] and [C 2| are single points.) Now sup- pose that the result is true for k and let C 1 have cardinality k+1. Let Bn be a convex linear n-cell in (2’1. Then {31 - Bn is a convex linear cell complex since Bn is principal in Cl. :2 - g(Bn) is also a cell complex since by (b) dim g(Bn) = dim Bn = n, and so g(Bn) is a principal cell in C 2. By the induction hypo— thesis there is a piecewise linear homeomorphism h: Cl-Bn|29-E2> [CZ-g(BnH sothat if AECl-‘Bn h:g(A) 2EE2> A. We will now extend h to int Bn. Note that since C 1 is a cell complex én C |C1 — Bnl and En =U n (C 0 En). CEQI'B Thus g[l§n] g( U n (C n Bn)) C€ g 1-B U n g(C 0 En) CE Cl-B U n [gm n g(Bnn C€Cl-B - (g(Bn))' . 15 Thus h: 6“ 9239s (g(Bn))' and we may extend h "cone- wise" [5] to a piecewise linear homeomorphism of Bn onto g(Bn). This is the desired homeomorphism. > on be a simplicial map. Lemma 3.6: Let f:|K[ If b denotes the barycenter of on and x 6 int (on), then f_1(x) is piecewise linearly homeomorphic to f-1(b). Proof: f_1(x) and f-1 (b) can be considered as convex linear cell complexes, the convex linear cells being given by f‘1(x) n c and f_1(b) n c for c e K. We will show that there is a one to one map g from the convex linear cells of f-1(b) onto those of f-1(x) such that g(AUB) =9(A) “g(B) and dim g(A) = dim A. The desired result will then follow from the previous lemma° Let 0 e K and suppose that o n f-1(b) # ¢. Since f is a simplicial map we must have f(0) = on and so 0 n f"1(x) # ¢. In the same way 0 fl f-1(x) # ¢ implies o n f_1(b) ¢ ¢. Thus if f_1(b) n ok ¢ ¢, f-1(b) n ok = Bk—n and f-1(x) 0 0k - Ck-n where Bk—n and Ck-n are convex linear cells of dimension k-n. Now define the mapping 9 as follows: g[f-1(b) 0 0k] = f-1(x) 0 0k . From the above discussion we have dim g [f-1(b) n ok] = dim [f_1(b) n 0k]. Moreover, 16 _1( guf'lao) n okl) n (f b) n okzn = gtf’Hb) n (ski n 0km k2) ) n (f-1(x) n o = f‘1(x) n (ok1 n o k1 I Corollary 3.7: Let f:|K{ L] be a simplicial map. f is strongly pointlike if and only if x e L implies f_1(x) is collapsible. Definition 3.1: Let |K| ‘E\\|K1| be an elementary simplicial collapse, (i.e. .K = K1 U (a * on) and Klfl(a * on) = a * on), and let K' be the subdivision of K obtained by starring from b, the barycenter of on. The simplicial map f:|K'| QE£2> |K1| defined by f(b) = a and f(V) = V for all vertices V e K', V # b will be called the simplicial map associated with the col- lapse |Ki ‘§\\|K1|. f; |K| onto Theorem 3.8: If > |L| is a strongly point— like map and |L|\\\_O, then [Ki‘\\\0. Proof: Since |L|\\§.O there is a triangulation L* of |L| such that |L*| \;\_O. The proof will be by in- duction on p , the number of simplexes in L*. 17 If p = 1 the result is trivial for then L* consists of a single vertex. Now suppose that the theorem is true for all p < k and let L* be a simplicial complex with cardinality k. By hypothesis IL*| \§§.O so there is a sequence of ele- mentary simplicial collapses ‘L*‘= lLol S\|L1| S\... b0 where L-)(- 2 _ .n L1 U (a * on) and L1 n (a * on) a * o . Note that [Lll ‘ENuO and L1 has cardinality k-2. Let 9=|L6I > [Lll be the simplicial map associated with the simplicial collapse [Lo] ';§\|Iq}. Subdivide -K to K' so that the map f:|K'| -—-> |L3| is simplicial. We will now show that the composition gf:|K'| > [L1| is strongly pointlike and use the induction hypothesis to conclude that IKI = IK'|\0- (i) If x e |L1| - a * 6n: then g-1(x) = x so f_lg-1(x) = f-1(x) which is collapsible by hypothesis. (ii) If x e a * 6n, g-1(x) is a line segment in a * on. Triangulate f-1(g-1(x)) so that f:f-1(g-1(xj) onto -1(x)'= 01 is a simplicial map. Then since f is strongly pointlike we may apply Theorem 3.4 to find that -1 —1 f (g (X))\0. , onto . . . Thus gf:|K | -—-—> L1 is strongly pOIntlike and the conclusion follows from the induction hypothesis. The following corollary and lemma will aid in the char- acterization of collapsible complexes. 18 Corollary 3.9: If the simplicial maps f and g are onto strongly pointlike where f:|K|'————> [LI and <3=l1‘|'(22't'9"> ‘ Pl, then the composition gf is strongly pointlike. Proof: Let x 6 (Pl. g (x) is collapsible since g .. ._ .31 — is strongly pointlike. Moreover f:f 1g (x) 22E2> g 1(x) _ -1 so by the above theorem f 1g (x)\\\_0 and gf is a strongly pointlike map. Lemma 3.10: If {R} ‘;\\]K1| is an elementary sim- onto> plicial collapse,the associated simplicial map f:[K'[ K1 is strongly pointlike. Proof: If x e 1K1} - (a * on) , f_1(x) - x. If x e a * on I f_1(x) is a polyhedral one cell. Thus in any case f-1(x)\\§.0. Theorem 3.11: If {Kl ‘;\\|L|, there is a strongly pointlike map f:|K*l onto [Ll where K* is a subdivision of K. Proof: Since IK] \;§_|L| there is a sequence of elementary simplicial collapses from K to L. IKI=IKol>s|K11>sm aslxplasm- Let fi:lKil onto IK. 1+1] be the simplicial map as" sociated with the elementary simplicial collapse |Ki| ”;$\|K1+1|. fp:|K;| 22£2> [L] is a strongly point~ like map by Lemma 3.10- Now subdivide K;_1 to K;_1 so that the map f : {K2_1{ O—M-Q p-1 > IK;| is simplicial. 19 Continue in this manner to arrive at the sequence of strongly pointlike maps f _ IK‘X-i = IKE+1I £L> 1K? __]'_> 1K5?) 1| >. 000 f f > [K Iii). |K2 |_.P;]:.> [Kl‘ _.E_.> IL|° p-2 p-l By Corollary 3.9 f = fpfp-1-'°' flfo is a strongly point~ like map. We are now able to state as a corollary the converse to Theorem 3.4. Corollary 3.12: If |K[\\\O, then there exists a onto> 1 strongly pointlike map f:|K*‘ o where K* is a subdivision of K. 2322;; Just note that if K is collapsible to a point the second last collapse takes K to a one-simplex. Remark: In view of Theorem 3.4 and Corollary 3.12 we have characterized collapsible polyhedra as those having a strongly pointlike map onto a one-simplex. SECTION IV FURTHER RESULTS CONCERNING COLLAPSIBLE COMPLEXES In this section we will use the results of section III to obtain necessary and sufficient conditions for a complex to be collapsible. Further, we will drive some properties of a collapsible complex embedded in a combinatorial manifold. The following two lemmas, along with the characteriza— tion of collapsible complexes of the previous section, will lead quickly to necessary and sufficient conditions that a complex be collapsible. onto Lemma 4.1: If f:|K| > (V0,V1> is a simplicial map and b is the barycenter of (V0,V1>, then there is a subdivision K' of K such that (i) K' is a first derived complex of K, onto (ii) f:|K'| -———e (V0,b> U is a simplicial map, and (iii) if o e K'|f‘1(b), then f(stK.(o))= U (b,v1>. Proof: We will star the simplexes of K in the order of decreasing dimension, ol,oz,...,os. If f:oi -—> Vj for j - 0,1, we will star oi from bi’ the barycenter of oi. If f:oi 2£E2> (V0,V1>, we will star Oi from a point b1 6 f_1(b) n int oi. Such a process yields a first derived complex of K. Call this K'. By [2] o e K' if an only if c is of the form 0 = where bik is the pOInt from which 0. is starred and o. > o. > ... > o. . 20 21 We first show that f:{K'| -—> (Vo,b> U is a sim licial ma . If f o. = V., ' = 0,1, th I f . = . p p ( lo) 3 3 en (01k) V3 for k = 0,1,...,n, and f(bi ) = Vj for all k. Thus k f o = V.. < > 3 If f(oio) = (vo,v1>, then either f(oi ) = (vo,v1> . k- . for all k = 0,1,...,n, or there is a least integer N < n such that f(oi ) = vj for a fixed j = 0,1 and all k k > N. In the first case f(bi ) 3 b for all k so f(0) = b. k In the second case f(oi ) = (V0,V1> for all k j_N, k . and f(o. ) = v. for k > N. Then f(b. ) = v. for all 1k 3 1k 3 k > N, and f(bi ) = b for all k i.N. Thus f(0) = . k . In any case f maps simplexes onto simplexes and f is a simplicial map. We now verify conclusion (iii). If f(0) = b, then G = , bi e oi and Oi > oi >...> oi . n k k 0 1 n By definition f(oi ) = (V0,V1> for all k. In particular k f(oi ) = (V0,V1>, so Oi has vertices uo and ul with n f(uo) = V0, f(ul) V1. Then by [2] the sequences 0. > o > ... > o. > uo and determine simplexes Co and C1 of K', . ' 000’ b0 IuO> and 11 1n C1 . 10 1.1 1n 22 Note that f(Co) = (Vo,b> and f(Cl) = . Then, since 0 < Co and o < C1, we have that Co U :1 C stK,(o) and f(stK,(o)) = (v0,b> U . Lemma 4.2: If f:{K| 9339+ is a simplicial map and b is the barycenter of (V0,V1>, then f-1(b) = le.(f-1(Vo)) where K' is the first derived subdivision of K described in Lemma 4.1. Proof: If 0 e K'|f-1(b), then f(o) - b so 0 fl f-1(V0) - ¢. By conclusion (iii) of Lemma 4.1 f(§t(o}) : (V0,b> U . Thus 0 < C where f(C) = 1 (Vo))° oh the other hand, if c e K'|le.(f‘1(vo)), then (V0,b> so C n f-1(Vo) # ¢ , and o e K']le.(f" 0 fl f-1(Vo) = ¢ and o < C e K' where fl 0 f"1(V0) # ¢. Since C n f_1(V¢) # ¢, f(C) = and f(o) C . Moreover, o n f-1(Vo) = ¢ so f(o) 0 (V0) = ¢ and f(0) = b. Thus 0 e f-1(b). Remark: Clearly it follows in the same way that f‘1(b) = 1k f‘1(v1)). K'( Theorem 4.3: Let |K|\\s0. Then there is a simplicial subdivision K* of |K| and subcomplexes A,B of K* with '(i) A,B full in K*, A # ¢, B # ¢, (ii) A n B = ¢, ' (iii) A U B :>(K*)0 (the zero skeleton of K*), (iv) |A|\O, |B|\O, and (v) 1kK.(A) = le.(B)\\§.O where K' is the first derived subdivision of K*. 23 Egggf: Since [K|\\e.0 we may employ Corollary 3.12 to find a simplicial subdivision K* of [KI and a strongly pointlike map f:|K*| 92£9> (VO,V1>. Let A = f-1(Vo), B = f_1(V1). Since f is a simplicial map, we note A and B are full in K? and since f is onto, A # ¢, B # ¢. Conclusions (ii) and (iii) are immedi- ate, and (iv) follows from the fact that f is strongly pointlike. Finally, by Lemma 4.2, le, (A) = le, (B) = f‘1(b)\o. Thus conclusion (v) follows. We will now prove a type of converse to Theorem 4.3. Theorem 4.4: Let K be a simplicial complex and A,B~ subcomplexes of K with (i) A,B full in K, (ii) A 0 B = ¢, (iii) A U B :)K0, (the zero skeleton of K), (iv) |A|\O, |B|\O, and (V) le,(A)\\§ 0 where K' is a first derived sub~ division of K. Then |K| \0. 22£2> (V0,V1> be the simplicial m: Let f: |K| map defined by f(A) = V0, f(B) = V1. Since A and B are full in K, f_1(Vo) = |A| and f-1(V1) = [B[. Thus, by hypothesis, f-1(Vo)\\§.0 and f_1(V1)\\§_0. By Lemma 4.2 f_1(b) = le,(A) so f-1(b)‘\ns0, and f is a strongly pointlike map. Then by Theorem 3.4 |K|\0. 24 Remark: Clearly, by symmetry, condition (iii) above can be replaced by le,(B)\\§~0. The next application of the results of Section III in- volves collapsible polyhedra embedded in a combinatorial manifold. Theorem 4.5: Let [Kl be a polyhedral n-complex in [m|, a combinatorial m-manifold. If [x] is collapsible, there exist two polyhedral m-cells N0 and N1 in [MI such that (i) if x e |K| we can choose N0 and N1 so that x e int N0, (ii) N0 U N1 3 |K|, (iii) No n [KI # ¢, N1 n [K[ # ¢, (iv) No n |K|\o, $11 n |K|\O, (v) N0 n |K|\0, N1 n |K[\0, and (vi) dim (hi n |K|)_<_n-1, i = 0,1. Proof: We will Suppose that K is a subcomplex of M. If not, there exist subdivisions .K1 and M1 of K and M respectively such that K1 C M1. Further, since )K1[\\§{0 there exists a subdivision .K2 of K1 such that [Kzl ‘;r\ 0. Extend the subdivision K2 of K1 to a subdivision M2 of M1. Moreover, if x e )K2|, we will assume that x is a vertex of K2. If not, we simply star .Mz from x and call the resulting triangulations M3 and K3. Note that since we arrived at »K3 by starring K2 and [K2] \;§~0I |K3| \§§.0, see [5]. In addition we will suppose that K3 25 is full in M3. If not, we have only to let M3' denote the first derived subdivision of M3 and K3' the cor- responding subdivision of K3. Then |K3'| \§§‘0 since it was obtained from K3 by starring. In order to save notation we will call K3', K and M3', M. In summation we now have ) KCM, b) x is a vertex of K, ) 'K is full in M, and d) [K] \s\0° Since |K| \;§~0, there exists a strongly pointlike map f:|K) 22£2> (V0,V1>. Since x is a vertex of K, either, f(x) = V0 or f(x) = V1. Assume that f(x) = V0. Star (V0,V1> from its barycenter, b, and subdivide K to Kf' SO that f: Kf' 22E2§ U is a simplicial map, and Kf' is a first derived complex of K (see Lemma 4.1). Extend .Kf' to a first derived subdivision of M, say M'. _ 1 . Now consider the first derived neighborhood of f (Vi), -1 I ' Z -1 . ' = N. Suppose that f(0) = V1, or f(0) = (b1,V1>, and o < C. Then since f is a simplicial map, f(C) 3 V1 or f(C) = . In either case C n f-1 o ¢ lkM,(f' we must have f(0) = b so f(y) = b and y e f-1(b). (V0) = 0 so C ¢ stM,(f_1(Vo)) and 1(Vo)). That is, 0 C No, a contradiction. Thus On the other hand, if y e f-1(b) then y e 0 e Kf' and f(o) = b. Since f(0) = b,‘ o n f-1(Vo) = ¢. But 0 C C e K where f(C) = (V0,V1> so 0 is the face of some 0' e Kf' Where f(o') = (Vo,b>. Thus f-1(VO) fl 0' # ¢. and o c lkM.(f‘1(vo)). Finally we will show that N0 0 |K| = f-1() and N1 0 [Kl = f‘1(). (2) Let y e No 0 |K[. Then y e No implies y 6 int 0, o e K; where o n f-1(Vo) # ¢. Thus f(0) = (V0) or f(o) - (V0,b>. In either case y e o C f-1(). On the other hand, if y e f_1(),then y e 0 e Kf' and either f(0) = (V0), f(0) = (Vo,b> , or f(o) - b. In the 27 first two cases 0 n f”1 (V0) # ¢. Thus 0 C stM,(f-1(Vo)) and o C No. If f(0) = b, o C f-1(b) and, by (1), o C No. The proof that N1 0 [K[ = f-1 is analogous. We are now in a position to verify that conclusions (i) through (vi) follow. (i) follows immediately from the construction since x e f-1(Vo) C int N0. (ii) follows since [K[ c f‘1() U f‘1() -N0 UNl. (iii) follows since f:[K[ > (V0,V1> was onto. _ . _1 (iv) follows from (1) above. That is, Ni 0 [K[ = f (b), -1 and f (b) is collapsible since f is strongly pointlike. “1 (v) follows from (2). That is, No 0 [K[ = f () and N1 0 [K[ = f-1(), and the fact that f is strongly pointlike. (vi) follows from the fact that f is a simplicial . . -1 __ map f: [K [ onto . SECTION V CONTRACTIBLE COMPLEXES WHOSE PRODUCT WITH THE UNIT INTERVAL IS COLLAPSIBLE The dunce hat D is obtained from the 2-simplex by identifying all three sides = = . D is of interest since it is one of the simplest contractible polyhedra which is not collapsible (there is no free face from which to start the collapsing). How- ever, it is well known, see [4], that D x I‘\§\0. This fact leads to the following conjecture. Conjecture 1: If K is a contractible 2-complex, then K x I \0. This conjecture is of particular interest since it implies the three dimensional Poincare' Conjecture [4]. Definition 5.1: Let M be a compact polyhedral mani— fold with boundary. Define a spine K of M to be a sub~ polyhedron 'such that M\K. By [4] we may assume that if K is a spine of M, then (i) K c int M, and (ii) dim K < dim M. The proof that Conjecture 1 implies the Poincare' Conjecture depends upon the following proposition (see, for example, [4]). PrOposition 5.1: Let M3 be a 3-manifold with a 2- sphere boundary and spine K2.28Then if K2 x I‘\§[0, M3 ‘ < 29 is a 3-ball. In this section we will apply the results of Section III to find a class of complexes K with the property that [K x I[\\§[O. Lemma 5.2: Let K be a complex with a subcomplex L. Then Kx1[\[(Kx {0}) U (L XI)[. Proof: Order the simplexes C with int C CZ[K[ _ [L[ in the order of decreasing dimension, Q1,C2,...,CS. First, [K x I|\[K x {0}[ U[([K[-int C1) x I] by collapsing the polyhedral ball C1 X I across its free face Q1 x [1]. In general, j-l' -. [K x{0}[U[ ([K[-'U int Ci)XI] \ [Kx{0}[ U 1-1 “[K[-,6 int Ci) x I] i=1 by collapsing Cj X I across Cj X [1). NOte that Cj X {1} is a free face of Cj X I 'since if Cj X [1] Ck Cj < Ck. Since the simplexes were ordered in the order of decreasing dimension, k < j, so X I, for j # k, then Cj X [1] < Ck X [1] and so j-l ' Ck x {1} ct [K x {0}[ U [([K[-U int Ci) x I]. i=1 Thus S [K> (V0,V1> by f((A U [L[) x {0}) v0, f((B U [L[) x {1}) V1, and linearly on [L[ X I. Note that since (A U [L[) X [0} and (B U [L[) X [1] are full in M, f‘1(v0) = (A U [L[) x {0}, and 31 f"1(v1) - (B U [L[)X {1}. Then since by hypothesis A U [L[\\§,0 and B U [L[\\§_O, we have f_1(Vo)\0 and f-1(V1)\0. Moreover, since f([L[ X [0]) = V0 and f([L[ X {1}) = V1, and f is de— fined linearly on [L[ X I, we have far;- V0 +%V1) = [L[ x [it-L Since [L[ X {%J is piecewise linearly homeomorphic to [L[, f-1(%' V0 +%V1)\ 0. Thus f is a strongly pointlike map, and so M‘\Q\0° Therefore, .K X I\M\0. Example 1: In Figure 3 we picture a 2—dimensional polyhedron, K, (the house with two rooms), due to R. H. Bing. Figure 3. 32 Although K is not collapsible, since it has no free faces, an application of Theorem 5.3 will show that K X I \\\0. Pass a plane P through K so that P con- tains therectangular disks D1 and D2 of Figure 3. This plane separates K into two components A and B. Figure 4 pictures A U (P‘fl K). B U (P n K) is similar. Figure 4. Clearly L = P nK\0, (A U L\0, and B U L\0. Thus, by Theorem 5.3, K XII\\\(). Example 2: Let (V0,V1,V2> be a 2-simplex and let V1' and V2' be two interior points of (V0,V1,V2> not 'co-linear with V0. Let Ki’ i - 1,2, indicate two c0pies ofi-the Space obtained by identifying the interVals = (V0,V1'>, (V0,V2> = : and let Li 33 denote the image of (V1,V2> in Ki° The polyhedron Ki is pictured in Figure 5. V0 Figure 5. Let K = K1 U K2 where the union identifies corre- sponding points of L1 and L2. Note that although .K is not collapsible, L1‘\§.0 and L1 separates K. If we let .K - L1 = A U B, then A U L1 = K1\0 and B U L1 = K2\0. Thus, by Theorem 5.3, K x I\0. Using the usual terminology we will call a counter- example to the 3-dimensional Poincare' Conjecture a fake 3-sphere (if such exists). If we triangulate this fake Sphere and remove the interior of a 3-simp1ex,the result- ing manifold with boundary will be called a fake 3-ball. Note that a fake 3-ball has a 2—sphere boundary. We may now prove the following theorem. Theorem 5.4: If K is a spine of a fake 3-ball, and T a tree in K which separates K into components 51 and 82, then either 81 U T or 52 U T is not collapsible. 34 2522:: Let M3 be a fake 3-ball with K as a spine. We may assume that T is a subcomplex in a triangulation of K. If 51 U T and 82 U T are collapsible we may apply Theorem 5.3 to find that K XII‘\m\0. Then by Propo- sition 5.1 M3 is a 3-ball. In the remainder of this section we will consider an 'additional method of collapsing K X I where K is a contractible polyhedron. Theorem 5.5: If K1\\§_0 and K1\\§[K by an elemen- tary collapse, then K X I\\§~0. 2522:; Let K1 = K U Bn, and Bn n K - Bn c B where Bn and B 1 are polyhedral n and n—1 balls reSpectively. By Lemma 5.2 K x I\(K x {0}) U (Bn' But (K X [0]) U (En—1 X I) is piecewise linearly homeo— 1 X I). morphic to K1. Let h denote the natural piecewise linear homeomorphism h:K X [0} > K. Then h is defined on -1 .. -1 . -1 . the n-1 cell Bn X [0] C (Bn X I) onto Bn CZBn. Thus by [5] h can be extended to a piecewise linear homeomorphism onto —-1 g:(K x {0}) U (Bn _x I) K U Bn 2 K1. -1 Thus, since K1\0, (K x {0}) U (Bn x I)\0, and we have 35 K x I\(Kx {0}) U (En-1 x I)\0. Example 3: Consider the dunce hat D. It is well known that D X I‘\§.0, [4]. However, the following ap— plication of Theorem 5.5 seems to be a somewhat easier way to prove that D X I\\\(). In Figure 6 we picture a two simplex, two of whose sides have been identified. The identification of a gen- erator of the cone with its base, as indicated by the num— bering of vertices in Figure 6, yields the dunce hat. Figure 6. Now expand D to the complex K indicated in Figure 7. K is simply D U B3 where B3 is the tetra- hedron with vertices v0,v1,v3,v4. Note that K\D since we may collapse B3 across the 2-simplex (V1,V3,V4>. Moreover K\\§.O as is indicated in Figure 8. The first 36 Vb .__.)_. -._._ _. .... .- lg‘ l Va. is /g§. / \\\ / / \ / V6 Figure 7. collapse pictured in Figure 8 is the collapse of B3 across the 2-simplex (V0,V1,V3>. In the second we collapse the 2-cell (V0,V1,V4> U (VO,V3,V4> across the 1-cell (V0,V3>. The third collapse collapses the 2-cell with vertices « V0,V2,V4,V1 across the one cell (V0,V2>. Finally we cole lapse the 2-cell with vertices V¥,V2,V4,V3 across the one cell (V4,V2> U (V2,V1>. The resultant complex pic~ tured in Figure 8 is‘a disk which is clearly collapsible. Thus by Theorem 5.5 D X I \nto. Corollary 5.6: If K\0 and K\L, then there exists an integer p such that L X Ip\\\u0. Proof: Let K\K1 \K2\... \Kp = L be a sequence of elementary collapses. The proof will be by induction on p. 37 Figure 8. 38 If p = 1 this is just Theorem 5.5. p—l Now su ose K K and K X I 0. pp p-1\\‘ .p p-1 ‘\\‘ Let K = K U B and P“1 P -1 . -1 Kp fl Bn = Bn C Bn where Bn and Bn are polyhedral balls. Then since _1 _ _. Kp_1 x Ip = (Kp x Ip 1) U (Bn x Ip 1) and -1 _1 m1 . n X Ip C(Bn x Ip ) -1 —1 (Kp x Ip ) n (Bn x Ip ) = B we have p—l p—l Kp_1 X I ‘\‘\Kp X I by an elementary collapse. Now apply Theorem 5.5 and we find that Kp X Ip\\m\0. Corollary 5.7: If K is a homotopically trivial poly~ hedron, then there is an integer p such that K X Ip‘\\\0. Proof: By [3] K is of the same simple homotopy type as a point. Thus there exists a complex L such that L \nsO and L\\apK. [We may now apply Corollary 5.6 to get the desired result. BIBLIOGRAPHY Tatsuo Homma, Piecewise linear approximations of em- beddings of manifolds (mimeographed notes), Florida State University, 1965. L. S. Pontryagin, Foundations of Combinatorial Top- ology, Graylock Press, 1952. J. H. C. Whitehead, Simplicial spaces, nuclei and m— groups, Proc. London Math. Soc., 45(1939), 243-327. E. C. Zeeman, On the dunce hat, TOpology, 2(1963), 341-358. E. C. Zeeman, Seminar on combinatorial topology (mimeographed notes), Institut des Hautes Etudes Scientifiques, Paris, 1963. 39 [[|[l[[lll 64 [[1 " W.[ M[ U" E“[ H u 7 o 3 o 3 9 2 4| 3 [NIH